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    "## 多层感知机\n",
    "\n",
    "多层感知机是在单层神经网络的基础上引入一到多个隐藏层，隐藏层位于输入层和输出层之间。 \n",
    "假设多层感知机的输入个数为$d$，只有一个隐藏层，其中隐藏层的单元个数为$h$，记隐藏层的输出为$H$，隐藏层的权重参数和偏差分别为$W_{d\\times h}$和$b_h$。只有一个隐藏层的多层感知机，其输出$O$为\n",
    "$$\n",
    "\\begin{aligned}\n",
    "H &= X W_h + b_h \\\\\n",
    "O &= H W_o + b_o\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "将上面两个式子联合起来，得到\n",
    "$$\n",
    "O = (XW_h + b_h)W_o + b_o = XW_hW_o + b_hW_o + b_o\n",
    "$$\n",
    "\n",
    "从联立后的式子可以看出，**虽然引入了隐藏层，却依然等价于一个单层的神经网络：其输出层的权重为$W_hW_o$。**而且，即便添加再多的隐藏层，以上设计依然只能与仅含输出层的单层神经网络等价。\n",
    "\n",
    "## 激活函数\n",
    "上述问题的根源在于全连接层只是对数据进行**仿射变换 affine transformation**，而多个仿射变换的叠加仍然是一个仿射变换。 解决该问题的一个方法是**引入非线性变换**，例如对隐藏 变量使用按元素的非线性变换，再作为下一个全连接层的输入。 这个非线性的函数被称为**激活函数 Activation Function**。\n",
    "\n",
    "### ReLU函数\n",
    "ReLU(rectified linear unit)函数提供了一个简单的非线性函数，给定元素$x$，函数定义为：\n",
    "$$\n",
    "ReLU(x) = max(x,0)\n",
    "$$\n",
    "可以看出，该函数之保留正数元素，并将负数清零。 "
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "import torch\n",
    "import numpy as np \n",
    "import matplotlib.pylab as plt\n",
    "\n",
    "def xyplot(x_vals,y_vals,name):\n",
    "    \n",
    "    plt.plot(x_vals.detach().numpy(),y_vals.detach().numpy())\n",
    "    plt.xlabel('x')\n",
    "    plt.ylabel(name + '(x)')\n",
    "    plt.show()\n",
    "\n",
    "x = torch.arange(-8.0,8.0,0.1,requires_grad=True)\n",
    "y = x.relu()\n",
    "xyplot(x,y,'relu')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "显然，当输入为负数时，ReLU函数的导数为0；当输入为正数时，ReLU函数的导数为1.在输入为0，ReLU是不可导的，我们取此处的导数为0，ReLU函数的导数图像"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "y.sum().backward()\n",
    "xyplot(x,x.grad,'grad of relu')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sigmoid函数\n",
    "\n",
    "sigmoid函数可以将元素的值变换到0和1之间：\n",
    "$$\n",
    "sigmoid(x) = \\frac{1}{1 + \\exp(-x)}\n",
    "$$\n",
    "sigmoid函数在早期的神经网络中较为普遍，但它目前逐渐被更简单的ReLU函数取代。在后面“循环神经网络”一章中我们会介绍如何利用它值域在0到1之间这一特性来控制信息在神经网络中的流动。下面绘制了sigmoid函数。当输入接近0时，sigmoid函数接近线性变换.\n",
    "\n",
    "sigmoid函数的主要问题在函数值过大或者过小时，梯度值太小，收敛缓慢。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "y = x.sigmoid()\n",
    "xyplot(x,y,'sigmoid')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sigmoid函数的导数为\n",
    "$$\n",
    "sigmoid'(x) = sigmoid(x) (1 - sigmoid(x))\n",
    "$$\n",
    "当输入为0时，sigmoid函数的导数达到最大值0.25；当输入越偏离0时，sigmoid的导数越接近于0。 梯度值太小，会导致收敛缓慢。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "x.grad.zero_() # 清零之前的梯度\n",
    "y.sum().backward()\n",
    "xyplot(x,x.grad,'grad of sigmoid')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### tanh函数\n",
    "\n",
    "tanh（双正切函数）可以将元素的值变换到-1和1之间\n",
    "$$\n",
    "tanh(x) = \\frac{1-\\exp(-2x)}{1 + \\exp(-2x)}\n",
    "$$\n",
    "当输入接近0时，tanh函数接近线性变换。虽然该函数的形状和sigmoid函数的形状很像，但tanh函数在坐标系的原点上对称。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "y = x.tanh()\n",
    "xyplot(x,y,'tanh')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "当输入为0时，tanh的导数达到最大值；当输入越偏离0时，tanh函数的导数越接近于0，这时候就收敛缓慢。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "x.grad.zero_()\n",
    "y.sum().backward()\n",
    "xyplot(x,x.grad,'grad of tanh')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 总结\n",
    "\n",
    "多层感知机是在输入层和输出层之间加入一个或者多个全连接隐藏层，并通过激活函数对隐藏层的输出进行非线性变换。 以只有一个隐藏层为例，多层感知机按照以下方式进行输出:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "H &= \\sigma(XW_h + b_h) \\\\\n",
    "O &= HW_o + b_o\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "在分类问题中，可以对输出$O$做softmax运算，使用交叉熵作为损失函数。在回归问题中，输出层节点个数为1，可以将$O$直接在线性回归中使用平方损失函数。"
   ]
  }
 ],
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.10-final"
  },
  "orig_nbformat": 2,
  "kernelspec": {
   "name": "python3",
   "display_name": "Python 3"
  }
 },
 "nbformat": 4,
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}